CMe) 
§ 2. I now determine the order of the scroll B formed by the 
principal tangents cutting $? in points 6 of the plane @. 
Out of each point B of the section 8" start (n— 3) (n? + 2) 
principal tangents; this number indicates at the same time the 
number of sheets of B which cut each other along gr". The inflect- 
ional tangents lying in ~” evidently belong (— 8)-times to the 
indicated scroll. So its order is equal to 
3) (n? + 2) + 3n (n — 2) (n — 8) = n(n — 1) (n — 3) (n + 4). 
According to § 1 (38n*—4n—6) principal tangents have their 
point of contact A on a” and one of their points of intersection 
B on Br. So this number indicates the order of the curve along 
which @* is osculated by B. Beside this curve of contact and the 
manyfold curve #8” the surfaces ¢" and B have still in common the 
locus of the points B’ which determine the principal tangents AB 
moreover on 9”. This curve (B) is of order n° (n—1) (n—3) (n-+-4) — 
—3n (Bn? — An — 6) — n(n — 3) (nV? + 2) =n (n — 2) (n—A4) (n° ++5n-+8). 
n (n 
§ 8. To find how often the point A coincides with one of the 
(1 — 4) points B’, I shall project the pairs of points (A, B) out of 
a right line /. The planes through / are arranged in this way in a 
correspondence with the characteristic numbers 7 (82?—4 n—6) (n—4) 
and 7 (n — 2) (n — 4) (n? + 5n-+8). Each right line a resting on / 
evidently contains (7—4) pairs (A, B), so it furnishes an (7 — 4)-fold 
coincidence. The remaining coincidences originate from coincidences 
A= B. Now n (8n?—4 n—6) (n—4) + n (n—2) (n—A4) (n? +5 nt3) 
n (n—1) (n—38) (n-++-4) (n—4) = n (n—4) (6n?+2n—24). So this is 
the number of fourpointed tangents which cut 9” in a point B of pr. 
The points of intersection of &” with its fourpointed tangents form 
a curve of order 2n (n — 4) (Bn? + n — 12). 
If / is the order of the scroll of the fourpointed tangents then it 
is evident that we have the relation 
nf = dn (lln —24) + 2n(n — 4) (Bn? + n—12) = 2n? (n— 3) (Bn—2). 
The fourpointed tangents form a scroll of order 2n(n—3)(8n—2). 
If we make the point of contact /’ of a fourpointed tangent to 
correspond to the (7 — 4) points G which that tangent has still in 
common with 9”, a system of pairs of points (/’, G) is formed, of 
which the number of coincidences can be determined again with the 
aid of the correspondence in which they arrange the planes through 
an axis /. By the way indicated above we find for this number : 
n(1in—24) (n—4)+2n(n—A) (3n?-+-n—12 )—2n(n—28) (8n—2) (n—4)= 
n(n— 4) (85n—60). 
The surface P* possesses 5n (i — 4) (In —12) fivepointed tangents. 
49* 
